Start from the simplest approach, you need a pump, a heat load, a heat dump (radiator) and an expansion valve to drop the pressure,
The cooling effect you get is from a pressure drop initiating a phase change, if the pressure drop is enough that it wants to change phase then you also get a boost from the latent heat of the phase change,
When a material reaches a point where it wants to change phase, generally it involves energy from the outside, e.g. water becoming steam needs additional energy from its surroundings, and vice versa when it condenses it gives off a similar amount of energy, this is how your refrigerator works.
So you design your loop around this phase change, just pressure vs temperature, pick a chemical and you can likely find its temperature / pressure phase chart online. So you need a pump that can reach a pressure where it becomes a liquid at your desired temperature range. but is a gas when not under operating pressure.
Most refrigeration loops do not use pressure feedback, only temperature feedback, run the pump for X time, then start to crack the expansion valve, you should see your heat load start cooling quite rapidly, then regulate the valve position to balance how much you want the inside to be cool, vs not heating up the radiator so much your fluid boils back to a gas, or so cold on the valve that it freezes over (that valve will be the coldest part of your loop in general)